By (REF ), any irreducible normal representation of \(W^*(G\times G)\)  is unitarily equivalent to \(\pi _{U_{z_1}}\otimes \pi _{U_{z_2}} : W^*(G\times G) \curvearrowright \mathcal {H}_{U_{z_1}}\otimes \mathcal {H}_{U_{z_2}}\) . By the final assertion of Proposition REF  \(z_1\)  must be \(\bar{z}_2\) , and hence \(\pi _{U_{\bar{z}_2}}\otimes \pi _{U_{z_2}}\) . Consequently, any irreducible spherical unitary representation of \(G<G\times G\)  must be of the form \(\pi _{\bar{U}}\otimes \pi _U : W^*(G\times G) \curvearrowright \mathcal {H}_{\bar{U}}\otimes \mathcal {H}_U\) . Using [1]} through Proposition 4.3(iii) we see that any spherical vector \(\xi \)  is unique up to scalar multiple. Hence, thanks to Proposition REF (ii) and [2]} with the notation \(T^\vee \)  there, it suffices to confirm that \((T^\vee \otimes 1)R_U = (1\otimes T)R_U\)  holds for any \(T \in B(\mathcal {H}_U)\) . However, this identity is known to be the characterization of map \(T \mapsto T^\vee \) ; see the proof of [2]}. Hence we are done.
